Optimal. Leaf size=32 \[ \frac {x}{\log \left (\frac {x^2}{4+\frac {3-x \left (x+\frac {x}{e^6}\right )}{2 x}}\right )} \]
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Rubi [F] time = 2.15, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-x^2+e^6 \left (9+16 x-x^2\right )+\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log ^2\left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2-e^6 \left (9+16 x-x^2\right )-\left (x^2+e^6 \left (-3-8 x+x^2\right )\right ) \log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\int \left (\frac {-9 e^6-16 e^6 x+\left (1+e^6\right ) x^2}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx\\ &=\int \frac {-9 e^6-16 e^6 x+\left (1+e^6\right ) x^2}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\int \left (-\frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {2 e^6 (-3-4 x)}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\left (2 e^6\right ) \int \frac {-3-4 x}{\left (3 e^6+8 e^6 x-\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\left (2 e^6\right ) \int \left (\frac {3}{\left (-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {4 x}{\left (-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\left (6 e^6\right ) \int \frac {1}{\left (-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\left (8 e^6\right ) \int \frac {x}{\left (-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\left (6 e^6\right ) \int \left (\frac {1+e^6}{e^3 \sqrt {3+19 e^6} \left (8 e^6-2 e^3 \sqrt {3+19 e^6}-2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {1+e^6}{e^3 \sqrt {3+19 e^6} \left (-8 e^6-2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx+\left (8 e^6\right ) \int \left (\frac {1+\frac {4 e^3}{\sqrt {3+19 e^6}}}{\left (-8 e^6-2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}+\frac {1-\frac {4 e^3}{\sqrt {3+19 e^6}}}{\left (-8 e^6+2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )}\right ) \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ &=\frac {\left (6 e^3 \left (1+e^6\right )\right ) \int \frac {1}{\left (8 e^6-2 e^3 \sqrt {3+19 e^6}-2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx}{\sqrt {3+19 e^6}}+\frac {\left (6 e^3 \left (1+e^6\right )\right ) \int \frac {1}{\left (-8 e^6-2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx}{\sqrt {3+19 e^6}}+\left (8 e^6 \left (1-\frac {4 e^3}{\sqrt {3+19 e^6}}\right )\right ) \int \frac {1}{\left (-8 e^6+2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\left (8 e^6 \left (1+\frac {4 e^3}{\sqrt {3+19 e^6}}\right )\right ) \int \frac {1}{\left (-8 e^6-2 e^3 \sqrt {3+19 e^6}+2 \left (1+e^6\right ) x\right ) \log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx+\int \frac {1}{\log \left (-\frac {2 e^6 x^3}{-3 e^6-8 e^6 x+\left (1+e^6\right ) x^2}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 31, normalized size = 0.97 \begin {gather*} \frac {x}{\log \left (-\frac {2 e^6 x^3}{x^2+e^6 \left (-3-8 x+x^2\right )}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 29, normalized size = 0.91 \begin {gather*} \frac {x}{\log \left (-\frac {2 \, x^{3} e^{6}}{x^{2} + {\left (x^{2} - 8 \, x - 3\right )} e^{6}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 33, normalized size = 1.03 \begin {gather*} \frac {x}{\log \left (-\frac {2 \, x^{3}}{x^{2} e^{6} + x^{2} - 8 \, x e^{6} - 3 \, e^{6}}\right ) + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 30, normalized size = 0.94
method | result | size |
risch | \(\frac {x}{\ln \left (-\frac {2 x^{3} {\mathrm e}^{6}}{\left (x^{2}-8 x -3\right ) {\mathrm e}^{6}+x^{2}}\right )}\) | \(30\) |
norman | \(\frac {x}{\ln \left (-\frac {2 x^{3} {\mathrm e}^{6}}{\left (x^{2}-8 x -3\right ) {\mathrm e}^{6}+x^{2}}\right )}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 34, normalized size = 1.06 \begin {gather*} \frac {x}{\log \relax (2) - \log \left (-x^{2} {\left (e^{6} + 1\right )} + 8 \, x e^{6} + 3 \, e^{6}\right ) + 3 \, \log \relax (x) + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.63, size = 77, normalized size = 2.41 \begin {gather*} \frac {x+x\,{\mathrm {e}}^6-8\,\ln \left (\frac {2\,x^3\,{\mathrm {e}}^6}{{\mathrm {e}}^6\,\left (-x^2+8\,x+3\right )-x^2}\right )\,{\mathrm {e}}^6}{\ln \left (\frac {2\,x^3\,{\mathrm {e}}^6}{{\mathrm {e}}^6\,\left (-x^2+8\,x+3\right )-x^2}\right )\,\left ({\mathrm {e}}^6+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 27, normalized size = 0.84 \begin {gather*} \frac {x}{\log {\left (- \frac {2 x^{3} e^{6}}{x^{2} + \left (x^{2} - 8 x - 3\right ) e^{6}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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